Using Stocking or Harvesting to Reverse Period-Doubling ... · competition and predation in different regions of the same phase space. Yakubu [16] shows that stocking or harvesting

J o u r n a l o f D i f f e r e n c e E q u a t i o n s a n d A p p l i c a t i o n s ,1998, Vol. 4, pp. 163-183R e p r i n t s a v a i l a b l e d i r e c t l y f r o m t h e p u b l i s h e rP h o t o c o p y i n g p e r m i t t e d b y l i c e n s e o n l y

0 1 9 9 8 O P A ( O v e r s e a s P u b l i s h e r s A s s o c i a t i o n ) N . V .P u b l i s h e d b y l i c e n s e u n d e r

t h e G o r d o n a n d B r e a c h S c i e n c eP u b l i s h e r s i m p r i n t .

P r i n t e d i n I n d i a .

Using Stocking or Harvesting to ReversePeriod-Doubling Bifurcations in DiscretePopulation Models*JAMES F. SELGRADE+

Department of Mathematics, North Carolina State University, Raleigh, NC27695-8205

(Received 1 August 1996; In final form 26 December 7996)

This study considers a general class of 2-dimensional, discrete population models where each percapita transition function (fitness) depends on a linear combination of the densities of the interactingpopulations. The fitness functions are either monotone decreasing functions (pioneer fitnesses) orone-humped functions (climax fitnesses). Four sets of necessary inequality conditions are derivedwhich guarantee generically that an equilibrium loses stability through a period-doubling bifurcationwith respect to the pioneer self-crowding parameter. A stocking or harvesting term which is propor-tional to the pioneer density is introduced into the system. Conditions are determined under whichthis stocking or harvesting will reverse the bifurcation and restabilize the equilibrium. A numericalexample illustrates how pioneer stocking can be used to reverse a period-doubling cascade and tomaintain the system at any attracting cycle along the cascade.

Keywords: Discrete models; period-doubling bifurcations and reversals; pioneer and climax popula-t i o n s .

Classification Categories: 39A10,39A12,92D25

1. INTRODUCTION

Nonlinear difference equations and ordinary differential equations are used tomodel the effects of the populat ion densi t ies of animals or plants on ecosystems.These models of population interactions possess complex dynamical behavior;even a single, discretely reproducing populat ion with a quadratic transi t ion func-

* Research partially supported by a grant from the British Columbia Forest Renewal Plan ResearchProject No: OPS. EN-109, British Columbia, Canada, and by the USDA-Forest Service, SouthernResearch Station, Southern Institute of Forest Genetics, Saucier, MS.

? Email: [email protected]. Tel: (919) 515 8589.

1 6 3

1 6 4 JAMES F. SELGRADE

tion exhibits a cascade of period-doubling bifurcations culminating in chaoticoscillations as illustrated by R. M. May [l]. Typically a period-doubling bifurca-tion occurs when varying a parameter of the system causes an eigenvalue of anequilibrium of the transition map (in the case of difference equations) or of thereturn map for a periodic solution (in the case of differential equations) to passthrough -1. For the difference equations, the equilibrium often loses stability anda stable cycle of period 2 appears; for the differential equations, the stable peri-odic solution becomes unstable and a stable solution of roughly twice its periodappears. Continued parameter changes may result in a cascade of period-dou-bling bifurcations and the onset of chaos. Franke and Yakubu [2,3] observe cas-cades of period-doubling bifurcations in discrete models for competitiveinteractions of pioneer and climax populations. For differential equation modelsof population interactions, Gardini et al. [4] illustrate a period-doubling transi-tion to a chaotic attractor for 3-dimensional Lotka-Volterra systems, and Bucha-nan and Selgrade [5] discuss similar behavior for a 3-dimensional model of theinteraction among pioneer and climax populations.

Usually chaotic behavior is undesirable in an ecological system. Here we studyhow systems which undergo destabilizing period-doubling bifurcations due tovariations in intrinsic parameters may be restabilized by extrinsic stocking orharvesting strategies. Buchanan and Selgrade [5] observe numerically that vary-ing a crowding parameter in a 3-dimensional interaction model with a periodicsolution produces a period-doubling cascade to a strange attractor. They showthat harvesting reverses this cascade and illustrate numerically that an appropri-ate level of harvesting may be chosen to maintain the system at any periodicattractor along the cascade. Thus the asymptotic behavior of this populationinteraction model may be controlled by harvesting. Unfortunately, obtaining ananalytical representation for the return map is not possible so a rigorous exami-nation of the bifurcations cannot be carried out in this example. This is a com-mon difficulty in the case of differential equations because it is not possible toobtain a simple surface transverse to the periodic solution where the return mapmay be analyzed. However, in the case of difference equations a rigorous mathe-matical analysis may be carried out. Here we restrict our attention to 2-dimen-sional difference equations which are analogues of the 3-dimensional ordinarydifferential equations studied by Buchanan and Selgrade [5]. Period-doubling forhigher dimensional discrete systems may be reduced to two dimensions, whereone dimension is the bifurcation direction and the other dimension includes allthe remaining hyperbolic directions.

We consider a general class of Kolmogorov models where each per capita tran-sition map (called the fitness) is a function of a linear combination of the densi-ties of the interacting populations. Previous studies of such systems include

REVERSE PERIOD-DOUBLING BIFURCATION 1 6 5

Comins and Hassell [6], Hassell and Comins [7], Hofbauer, Hutson, andJansen [8], Cushing [9,10], Selgrade and Namkoong [11,12], Franke and Yakubu[2,3,13,14,15], Yakubu [16], Buchanan and Selgrade [5,17], and Selgrade andRoberds [18,19]. One fitness function in our model will be a monotone decreas-ing function (pioneer fitness) and the other, a one-humped function (climax fit-ness). The interaction of a pioneer population and a climax population permitscompetition and predation in different regions of the same phase space. Yakubu[16] shows that stocking or harvesting may be used to obtain stable coexistencein a system of two competing pioneer populations where there is exclusionarydynamics without stocking or harvesting. Stone [20] discusses reversingperiod-doubling bifurcations in a model for a single pioneer population with aconstant immigration (stocking) term. Stone’s bifurcation parameter is the intrin-sic growth rate and the period-doubling reversals appear as “bubbles” in thebifurcation diagram where the population density is plotted against the growthrate. Our intention is to determine when period-doubling bifurcations can bereversed by varying the stocking or harvesting parameter which is extrinsic to thepopulation interaction. Bubbling in our setting will occur when the 2-parameterbifurcation curve has a critical point, and we will consider this possibility infuture studies.

Section 2 discusses some specific properties of the model equations and bio-logical rationale for these properties. In section 3 we derive four sets of neces-sary inequality conditions which also guarantee generically that an equilibriumloses stability through a period-doubling bifurcation with respect to an intrinsicpioneer self-crowding parameter. Section 4 introduces into the system a stockingor harvesting term which is proportional to the pioneer density. We determineconditions under which this stocking or harvesting will reverse the bifurcationand restabilize the equilibrium. In section 5 we show that if the pioneer fitnessfunction has nonnegative concavity, which is the case for all familiar examples inthe modeling literature, then an equilibrium of prey-predator type may be restabi-lized if and only if stocking is done. Section 6 studies the case of the equilibriumof competitive type. We discuss an example where the size of the concavity ofthe pioneer fitness function determines whether stocking or harvesting will resta-bilize the equilibrium. Small positive concavity requires harvesting and largepositive concavity requires stocking. Finally, we illustrate numerically how pio-neer stocking can be used to reverse a period-doubling cascade and to maintainthe system at any attracting cycle along the cascade. Since these results are local,they may be applied to any Kolmogorov system of difference equations whereeach per capita growth rate is a function of a linear combination of the populationdensities and where the bifurcating equilibrium is either of prey-predator type orof competitive type.


2. BACKGROUND AND MODEL EQUATIONS

In order to study period-doubling bifurcations we consider systems of 2-dimen-sional, nonlinear difference equation which model the interactions of discretelyreproducing populations. Let x1 and x2 denote the densities of two populations.Let y, for i = 1,2, denote the weighted total density variable for xi, i.e.,

Yi = cilxl+ ci92

where Cy 2 0 is called the interaction coefficient and reflects the effect of the j-thpopulation on the i-th population. The 2 x 2 matrix C = (cd) is called the interuc-tion matrix. The per capita growth rate, the fitness fi, of the i-th population is asmooth function of yi. Our model equations are

x; = x& (yi) i= 1,2 (2.1)

where xi denotes the density of the i-th population at the next generation.Because (2.1) is of Kolmogorov-type, if the values of fi are always nonnegativethen the nonnegative quadrant is invariant for solutions to (2.1). However, wewill not assume that this is always the case.

Introducing the weighted density variable, y, has the advantage of separatingthe i-th population’s response to density, A, from the competitive or cooperativeeffect of each population. Typically this response may be characterized bymonotonicity properties of the fitness f;: as a function of the weighted density yi.Because of the detrimental effects of crowding, h may be a decreasing functionof yi. Iffy is monotonically decreasing for all values Of y, (see Figure l), then xi isreferred to as a pioneer population. A variety of pioneer fitness functions appearsin the modeling literature. For instance, exponential (see Moran [21] and Ricker[22]), rational (see Hassell and Comins [7]), and linear (see Selgrade and Rob-erds [18,19]) functions are used. On the other hand, Allee [23] discusses manyexamples of the beneficial effects of increasing density on both reproduction andsurvival rates, especially at low density levels. In forest ecosystems, certain treepopulations such as oak and maple benefit from the presence of additional treeswhich provide protection and improved soil conditions. However, as total densityincreases, ultimately the adverse effects of crowding reduce individual fitness. Iffi monotonically increases up to a unique maximum value and then monotoni-cally decreases as a function of weighted total density, (see Figure l), then xi iscalled a climax population. Examples of climax fitnesses are exponential andquadratic functions, see Cushing [9,10] and Selgrade and Namkoong [12].Henceforth, we assume the x1 is a pioneer population and x2 is a climax popula-tion.


FIGURE 1 Pioneer and climax fitnesses

An equilibrium in the interior of the nonnegative quadrant occurs where all fit-nesses assume the value 1. Since fr is a pioneer fitness then we assume that thereis exactly one value y*1 > 0 so that fi (y? ) = 1. Hence the h-isocline is the line

Y*1 = cr+r + ~~9~. Also we assume this value y? is nondegenerate, i.e.,f;( yr ) z 0. We take the climax fitness f2 to have exactly two positive valueswhere i t assumes the value 1, which also are nondegenerate. Thus the isoclines ofthe climax population are two parallel lines. An equilibrium in the positive quad-rant occurs precisely where the pioneer isocline intersects one of the climax iso-clines. If y* = (yT,YS) is a vector such that h(yT ) = 1, for each i, then aninterior equilibrium E = (e,, e.J is a solution to the system of linear equations

Hence we find

CE=y+. (2.2)

e, = YT c22 - J? Cl2 ec11 -y:c21detC ’

e2 =detC ’ (2.3)

For E to be in the positive quadrant, both numerators in (2.3) must have thesame sign as detC. This equilibrium is isolated if detC # 0 , which we alwaysassume.


3. PERIOD-DOUBLING WITHOUT STOCKING OR HARVESTING

The stability of an interior equilibrium E may be determined by the Jacobianmatrix of the right side of (2.1). At x = (x1, x2) this matrix is given by:

(3.1)

When (3.1) is evaluated at E, the first matrix in (3.1) becomes the identity. LetD(E) denote the 2 x 2 diagonal matrix in (3.1) with entries eJ;(yI) ande&(yi) . Notice that the eigenvalues of the product D(E)C are left-translationsby the amount 1 of the eigenvalues of J(E). If the eigenvalues of J(E) are insidethe unit circle in the complex plane, which implies that E is asymptotically sta-ble, then the eigenvalues of D(E)C are inside the circle of radius 1 centered at - 1in the complex plane. Hence, the trace of D(E)C is between - 4 and 0 and thedeterminant of D(E)C is between 0 and 4, i.e.,

0 < det[D(E)CJ = ele& (yt)f;(y;) detC < 4.

Since fl(y;) is negative, for (3.2) to hold we must have

(3.2)

fh(yl) detC < 0. (3.3)

Notice that (3.2) and (3.3) still hold if one eigenvalue of J(E) is -1 and theother is inside the unit circle. If one eigenvalue of J(E) is -1 then the other eigen-value of J(E) not equal to 1 is equivalent to det[D(E)Cj # 0; if one eigenvalue ofJ(E) is -1 then the other eigenvalue of J(E) not equal to -1 is equivalent todet[D(E)C] z 4. Thus, for an equilibrium to lose stability by its smaller eigen-value passing through -1, it is necessary that (3.2) holds and that the slope of theclimax fitness f;(yi) and detC have opposite signs. The term detC measures thedifference between the intraspecific and the interspecific competition. If theintraspecific competition is less than the interspecific competition (detC c 0)then f;(y;) must be positive and, hence, the equilibrium E, must occur wherethe pioneer isocline intersects the climax isocline which is closer to the origin. Atsuch an equilibrium, the pioneer and climax populations interact Eke prey andpredator, respectively. We refer to El as the equilibrium of prey-predator type.On the other hand, if the intraspecific competition is greater than the interspecificcompetition then f;(yG) must be negative; so the bifurcation equilibrium E, isthe intersection of the pioneer isocline and the climax isocline farther from theorigin, and the populations truly compete with each other at this equilibrium. Werefer to E, as the equilibrium of competitive type.

REVERSE PERIOD-DOUBLING BIFURCATION 169

Computing the eigenvalues h, of J(E), we obtain

A, = fj (2+erf’r(y;) cl1 + e2.GW c22)

2; Kelf’dr;) cl1 - e2f$(A) c22)2 + 4ele2c12c21f’l(Y;)f;(Y~) Y2. (3.4)

From (3.4) observe that the eigenvalues of the equilibrium of competitive typeare real and distinct. However, the equilibrium of prey-predator type may havecomplex eigenvalues and, in fact, undergo Hopf bifurcation, see [ 181. But ourbifurcation equation, (3.5) below, will not be valid if the eigenvalues havenonzero imaginary parts. For a stable equilibrium to undergo period-doubling,we need h to pass through -1. Solving (3.4) for h = -1, we obtain

0 = 4 + 2ed’l(r;) cl1 + 2e2fh(y;) ~22 + ele2f’l(y;)fL(yl) detC. (3.5)

The right side of (3.5) may be rewritten as1 + tr J(E) + det J(E).

Clearly this sum is positive if the eigenvalues of E have nonzero imaginaryparts. Selgrade and Roberds [18,19] indicate that the intraspecific competitioncoefficients cI1 and cZ2 are convenient, intrinsic bifurcation parameters. For sim-plicity we choose cI1 as our bifurcation parameter and solve (3.5) for cI1 in termsof parameters which are independent of cm i.e., the other interaction coefficients,the total density variables yLF at equilibrium, and the slope of each fitness at theappropriate yi* :

n 4qy$l+ 2Y~C,,C,,filYy,*) +c21Yl*(Yl%- Y2*c12)fil(Yflh’(Y2*)c11= I

[2+ Y2*h(Y2)Ii 2% +f1I(Yi”)(YP,,- Y2*C12)) (3.6)

For the bifurcation to occur, it is necessary that the right side of (3.6) be positive.To determine the direction of bifurcation we need to find d?~- ldc,, at cl1 = &I .

From (2.3) we compute

% -c22e1

- = detCac11(3.7)

.We differentiate h with respect to cI1 and use (3.5) and (3.7) to evaluate at cI1 = CU.This is a tedious algebraic computation which begins by differentiating ?L in(3.4) yielding two terms, one of which has a square root in the denominator. Butwhen cI1 = &t and h = -1 then (3.4) may be used to replace this square rootterm by 14 + edi &t t ezf;(yl) cZ2]. Then a common denominator isobtained and (3.5) and (3.7) are used to simplify the expression giving

dh_- -2elc2J 2 + yEf;(Yl) 1dc11 - e,detC[4 + e,f/(y;)& + ed2’($)c221 ’

(3.8)


Using (3.2) and (3.4) we conclude that the term in brackets in the denominatorof (3.8) is positive. Hence the signs of detC and [2 + y;*f ‘,(y;)] determine thesign of (3.8). If (3.8) is negative then E loses stability as cl1 increases; and if(3.8) is posi t ive then E loses s tabi l i ty as cn decreases.

If the bifurcation occurs at the equilibrium El of prey-predator type, where theslope of the climax population fh(yl) is positive, then from (3.3) detC must benegative and so is the term y; cZ2 - yl cl2 because of (2.3). Thus each term inboth the numerator and the denominator of (3.6) is positive, and there is always apositive solution iii to (3.5) given by (3.6). In fact, at the value iii, a computa-tion shows that detC < 0; so assuming detC c 0 is extraneous. Also the numeratornof the second coordinate of El must be negative at cl1 . Substituting (3.6) into(2.3) and simplifying gives this numerator as

-24 2 +Y;f’1(Y;)l(YTc22-YIc12)y&1-yTc21 = [2 +ygf;(y;)]{2c,, t.f; (y;)(y;c22-y;4}’ (3.9)

Since the denominator in (3.9) is positive, we require that 2 t ylf’, (y;) c 0 sothat the fraction is negative. Hence, for a period-doubling bifurcation withrespect to the parameter cl1 at the equilibrium El, we must assume that

case (i): yrc,, -y;c,, < 0 a n d 2 +Y;f’l(Y;) <o. (3.10)

The inequalities in (3.10) guarantee that the numerators of the coordinates ofEl are negative and, hence, E, is in the positive quadrant when cl1 = iii . In addi-tion, the second inequality in (3.10) and the fact that detC c 0 imply that (3.8) isnegative. Thus E, loses stability as cl1 increases through iii. Figure 2 indicatesthe positions of the pioneer and climax isoclines in this case. The fact that thenumerators of the coordinates of El are negative results in the x,-intercept,y; /cri, of the pioneer isocline being to the right of the climax isocline and thex,-intercept of the pioneer isocline being below the climax isocline. As cl1increases, the x,-intercept of the pioneer isocline moves to the left which causesE, to slide down the climax isocline as E, loses stability. This agrees with thesigns of the derivatives in (3.7). Hence as the pioneer self-crowding parameter,cm increases, the equilibrium of prey-predator type moves down and to the rightin phase space and loses s tabi l i ty .

If the bifurcation occurs at the equilibrium E, of competitive type, where theslope of the climax population $(yi) is negative, then from (3.3) detC must bepositive and so are the numerators in (2.3). Thus the pioneer and the climax iso-clines interchange positions in Figure 2, i.e., see Figure 7. This implies thatperiod-doubling bifurcations with respect to the parameter cn may not occur in thesame system at both El and E2. Since &(y;) < 0, the numerator and the denomina-tor of (3.6) for iti may not agree in sign and, in fact , the denominator may be zero;and so the feasibil i ty of a period-doubling bifurcation with respect to the parameter


x2

FIGURE 2 Pioneer (p) and climax (c) isoclines and El

cn at E, requires addit ional assumptions. We consider cases depending on the signs

of P + YZ(YG)I and P + YX(Y;)I. If both P + YG.G(YG)I ad P + YX(YT)Iare positive then (3.9) is negative which precludes E2 from being in the positivequadrant .

If both [2 + YX(YG)I and P +YX(Y~I are negative then the numerator in(3.9) is positive and we need

2C**+f;(Y;)(Y;c22-Y;c12) <,o (3.11)

for the denominator in (3.9) to be positive. Also 211 > 0 follows from (3.11).Hence the following assumptions permit a bifurcation at E,,

case (ii): Y;c,,-Y;c,2 ’ 0, 2 + Yx(Y;) < 0, 2 +Y;f;(Y;) < 0,

and 2~22+f;(~;)CY1*~22-~;~12) CO. (3.12)

Given (3.12), dh- ldc,, > 0 so E2 loses stability as cl1 decreases.If [2 + y2;(y;)] c 0 and [2 + y;f;(y;)] > 0 then the inequality in (3.11) is

reversed, and so (3.9) is positive. But to get El1 > 0 we need to assume

4% + Y;~(Y~){2c22+f;(Y;)Cy;c,,-Y;cl,) > <a (3.13)

Thus the following conditions permit a bifurcation at E2,

case (iii): Y;c,,-Y;c,, ’ 0, 2 + YSXY;) < 0, 2 + Y;f;(Y;) ’ 09

and 4~12 + Y;~(Y~){~c,~+~;(Y;)~,;c~~-Y;*c~,) > CO. (3.14)

In this case dLMc,, < 0, so E2 loses stability as cl1 increases.

172 JAMESESELGRADE

If [2 + y;fi(y;)] > 0 and [2 t y;f;(yi) ] < 0 then we need to assume that theinequality in (3.11) is reversed so that (3.9) is positive and to assume that the ine-quality in (3.13) is reversed so that 211 > 0. Hence bifurcation occurs at E2 if,

case (iv): Y;c,,-Y;cl, ’ 0, 2 + YmY;) ’ 0, 2 + Y;f;(Y;) < 0,

4% + Y;uY;)Gc*2+f;(Y;) (Y~c,,-Y;d 1 ’ 0,

and%+fi(~;) (Y;c~~-Y~) ‘0. (3.15)

Here dh-/dcll > 0, so E, loses stability as cl1 decreases. For cases (ii), (iii), and(iv) computations show that detC > 0 at &t , which is required for E2 to be in thepositive quadrant.

The inequality conditions in cases (i) through (iv) guarantee that E loses stabil-ity as cl1 passes through 211 because the smaller eigenvalue passes through -1,which generically is a period-doubling bifurcation. However, it is much moredifficult to determine the shape of the curve of period-2 points. Conditions suffi-cient to prove that this curve is “parabolic” in shape are derived in [19] andinvolve combinations of the first three derivatives of the fitness functions.

The preceding discussion in section 3 establishes the following result:

PROPOSITION 3.1. Suppose that E is an equilibrium of (2.1) in the positivequadrant. Assume that

0 < wfiM)f b<y;> &Cc4and that one of the sets of inequalities (i), (ii), (iii), or (iv) holds. Then E losesstability as cl1 either increases or decreases through &I given by (3.6), becauseits smaller eigenvalue passes through -1. Its larger eigenvalue remains inside theunit circle.

4. DENSITY-DEPENDENT STOCKING AND HARVESTING

Since varying the pioneer self-crowding parameter cl1 destabilizes the equilib-rium, one would suspect that stocking or harvesting the pioneer population mayrestabilize the equilibrium. We investigate this possibility by adding a den-sity-dependent stocking (a > 0) or harvesting (a c 0) term to the pioneer differ-ence equation in (2.1) to get:

(SW

The amount of stocking or harvesting is directly proportional to the current pio-neer density with proportionality constant “a”. Mathematically, this is the sim-


.

p lest way to include stocking or harvest ing because (SH) retains the Kolmogorovform and the isoclines are still lines. However, from a practical point of view, amanager of an ecosystem would have to know the pioneer density to stock orharvest at a proportional rate. We consider (SH) as a system in the two parame-ters a and err.

An equilibrium E = (er, e2) in the positive quadrant is a solution of the simulta-neous equat ions:

.(4.1)

1 =fAc21e1 + cz24.The climax isoclines are the two parallel lines determined by the values of the

weighted density variable y, where fi(v2) = 1 as in the case when a = 0. The twovalues for y; such that f2(yG) = 1 are independent of err and a. The pioneer iso-cline is determined by the value of y, where f&J = 1 - a which we denote yI (a),see Figure 3.

01 YiW 3

FIGURE 3 Pioneer fitness fi with stocking (a > 0)

Since f&Q is decreasing, it is clear that y; (a) is an increasing function of a. Infact, dy; /da = -l/f ;(y;) > 0. Notice that yr (a) is independent of cll. The coordi-nates of E are still given by (2.3) except that y; is a function of a. From (2.3) wecompute

de1 -422 de2-=au f;(y;(a))detC and % = f;(yX$)defC. (4.2)

174 JAMES F. SELGRADE

FIGURE 4 Pioneer (p) and climax (c) isoclines near E, for a 2 0

The pioneer isocline is given by the linear equation

CllXl + Clzx2 = Y; (4. I (4.3)Since y; (a) is increasing as a increases, the x,-intercept and x,-intercept of this

isocline move away from the origin as a increases but its slope is always -c,,/c,,.In the case of the equilibrium El of prey-predator type, increasing a moves E, upand left along the climax isocline but increasing err moves E, down and right, seeFigure 4. The opposite behavior occurs at E2 because the positions of the iso-clines interchange, see Figure 7.

The derivative of the right side of (SH) evaluated at an equilibrium E is:

(4.4)

This matrix is analogous to (3.1) except that er, e2, and yI depend on theparameters err and a. A period-doubling occurs when the smaller eigenvalue h-in (3.4) passes through -1. Hence we need a solution (a,crr) to equation (3.5). Wedefine the bifurcation function G(u,crJ to be the right side of (3.5) and repeatthat equation here:

0=4+2ed;(y;(a)) c,,+%.fd(yl) c22+edf;(y;(a)) &‘(Y;) detC=Wd(4.5)

Each of the four inequality conditions (i), (ii), (iii), or (iv) guarantees that theequation G(a, err) = 0 has a solution (a,~,~) = (0, &t ) where &r is given by(3.6). Thus, near (0, &r ), the bifurcation curve BC is a nonempty set of points:


BC = {(a, cll) : G(a, cll) = 0).

.

If a = 0 and if the equilibrium E loses stability because of a variation in cir thenwe would like to determine if stocking or harvesting (a z 0) will restabilize E.Understanding the nature of the curve BC near (0, &i ) will answer this question.

We appeal to the implicit function theorem to show that curve BC may be con-sidered the graph of cri as a function of a near (0, iit) . The appropriate suffi-cient condition is

From (4.5) we compute dG/acll where Y*1 = Y?(O) :

dG = 2&l f;( yr > 2 + 2elfX rT 1 + 2kUY% ) ae2

%l dc+ 11

ae’e2fi(Yi: )fi(Y%.)detCac11

+ el$ f;( YT IfiCy* 1 detc + wle2 fi(rT )fl(~9 ). (4.6)

Inserting (3.7) into (4.6) and rearranging terms gives

aG- = SC [-2c12f; ( YT > e2 + 2 c22.G (Y*2)ac11

e2 +

ele2 JY(YT )NY*~ ) detcl.

From (4.5) we obtain

~c~~.f~( yyi )e2 + qe2 fi(yt )f;(~*2) detc = -4-2kfX~T ) el

and substitute into (4.7) to get

aG-=-532+Y:f;(yf)l.ac11

(4.7)

(4.8)

Using (2.3) for ei and e2 in (4.8) leads to the alternate form

aG [2+Y*2f;(Y*2)1{2c,,+f;(YT)(YTc,-Ylc,,)}-=%I detC (4.9)

The advantage of (4.9) is that the right side is easily computed from the inter-action coefficients and the weighted density variables. The sign of aG/acii, is

determined by the signs of detC and [2 ty? f;( yT ) 1, the latter is given in con-ditions (i) through (iv). For cases (i) and (iii), aG/acn is negative; for cases (ii)and (iv), aG/ac,, is positive. Hence we have established the following result:


PROPOSITION 4.1. Suppose that one of the sets of inequalities (i), (ii), (iii), or(iv) holds. Then the bi furcat ion curve BC in the (a,@space ofparameters is thegraph of cl1 as a function of a near thepoint (0,211).

The slope of this graph at (0, &r ) is the negative of dGlaa divided by aG/ac,,.Computing aGlaa from (4.5) and evaluating at (0, &r ), we obtain

aYT~=Gf;(yf 1% +2&elf’;(yT )a de2+ 2%zG(Y*, I= +

2 e&‘( y1* If2’(y2* 1 detC

+ elzae2 fi(yT )f~(y*2)detC+elezf;(yT )zf;(y*Z) detC. (4.10)

Inserting (4.2) into (4.10) and rearranging terms givesdG 1

aa =f ;( y? ) detCKc&( YT ) + eJ ‘;( YT ) detC){-2& -

+%f;.(Y*2) w22+f;(YT) (YT C22-Y%C12)11. (4.11)Using (3.6) to substitute for &t and (3.9) to substitute for y% &r - yr c2r, we

iset

-2&l -fi(Y?) (Y*2C11-YTC21)=2C-4c12c21

2 2+f;(yT)(y~c22-Y*2C12)’

and inserting this into (4.11) we obtain

aG-= czlf;(Y? I{ 2c22 +f i(Y*l )(Y*l c22-Yyi c12))aa fi ( Y: WC

%2c2Jc22fi( YT 1 + elf ‘i( YT ) de4

-fXyT WC{ 2c2, + f 3~7 NY? c~~-Y? c12)b

For each of our four cases, the sign of aG/aa may not be determined in general.However, reasonable biological assumptions will permit us to establish the signof (4.12) and to conclude appropriate stocking or harvesting strategies to restabi-lizing the equilibrium.

5. RESTABILIZING THE EQUILIBRIUM OF PREY-PREDATOR TYPE

The equilibrium El of prey-predator type loses stability as err increases through&r when the inequalities in (i) are satisfied. Since f;( y3 ) > 0 and detC c 0 at


El, it is clear that the first term in (4.12) is positive. For the second term in (4.12)to be positive we need that the term in brackets in the numerator is negative. Thisterm depends on the concavity of the pioneer fitness. For the pioneer fitnesses inthe modeling literature, ft” 2 0. For instance, the exponential and rational fit-nesses are concave up and the linear fitness has zero concavity. Hence, it is rea-sonable to assume that

f’;(YT ) 20 (5.1)From (5.1) it follows that (4.12) is positive. Since dG/dcrr c 0 in case (i), the

graph of the bifurcation curve BC has positive slope at (0,&r ), see Figure 5.At a point in parameter space (O,cr,) where err > &r, El is unstable. To restabi-

lize E, by varying a only, we must reach a point in parameter space below and tothe right of the bifurcation curve, i.e., stocking the pioneer restabilizes the equi-librium. Biologically this is somewhat counterintuitive. Since increasing pioneerself-crowding destabilizes E,, one might suspect that reducing the self-crowdingby harvesting would restabilize El. However, from Figure 5, it is clear that above(0, Err ) but near (0, &r ) harvesting will keep E, unstable. In fact, harvesting thepioneer in this system-increases pioneer density and decreases climax density atequilibrium. Thus the predatory effect of the climax population on the pioneer,population is reduced by harvesting; apparently, this predation is crucial to main-taining equilibrium stability. The geometry of Figure 4 may help in understandingthis phenomenon. With a = 0 as cl1 increases through &r, the pioneer isocline

01 a

FIGURE 5 Bifurcation curve near (0,211)


swings to the left and the equilibrium Et loses stability as it moves down the climaxisocline. As Et moves down the climax isocline, the pioneer density is increasingand the climax density is decreasing. Stocking or harvesting displaces the pioneerisocline parallel to itself (see Figure 4) - upward for stocking and downward forharvesting. Hence, harvesting would move El farther down the climax isocline.Stocking moves E, up the climax isocline to a larger climax density and a smallerpioneer density and restabilizes E,. The restabilization depends on factors moresubtle than the position of E,. It depends on a balance between the predatory effectof the climax on the pioneer and the pioneer self-crowding. This is reflected in theeigenvalues of the matrix J(E,) which vary with both a and cu, see (4.4). Thus, forall biologically reasonable fitness functions, the loss of stability of the equilibriumof prey-predator type via a period-doubling bifurcation with respect to cl1 may bereversed only by stocking. Specifically, we have obtained:

PROPOSITION 5.1. Assume that (i) holds and that f ;( y? ) 2 0. Then the equilib-rium E, of prey-predator type is restabilized if and only if stocking is done (i.e.,a > 0).

6. RESTABILIZING THE EQUILIBRIUM OF COMPETITIVE TYPE

For the equilibrium E, of competitive type to lose stability at ill, one of the con-ditions (ii), (iii), or (iv) must be satisfied. Since ~G/$r # 0, the bifurcation curveBC is still the graph of cl1 as a function a near (0, &t ); but its slope is more diffi-cult to determine. Here we illustrate that in some situations this slope is positiveand in other situations it is negative. Recall that at E, we have f b( y3 ) c 0 anddetC > 0. The fact that detC > 0 is pivotal to determining the sign of aG/&z. Thesign of the second term in (4.12) depends on the sign of the term in brackets inthe numerator. The first addend is negative but the second addend is nonnegativebecause typically f ‘;( yt ) 2 0. We present two examples of how the concavityof the pioneer fitness affects the slope of the bifurcation curve.

Consider the inequality conditions in (iv). First assume that the pioneer fitnessis linear so f ‘; = 0. By finding a common denominator, we combine the twoterms in (4.12) to get the numerator:

--2clzczlczzf;(Yf 112 + Y*2fh(YYi )I -C12C21Yqf;(Y*2)

[f;(Yt )12(Y;hCz2- Y?d * (6.1)

i

,


Each term in (6.1) is positive so the numerator of dG/da is positive, and thedenominator

KC YT 1 detCI2 C22+J?(YTNYTC22- Y&2)1) (6.2)

is negative. Hence, aG/aa < 0. From (4.8) or (4.9), we see that aG/dcr, > 0. Thusthe slope of the bifurcation curve is positive. Since (3.8) is positive, E2 loses sta-bility as err decreases through &r . In this case harvesting is needed to restabilizeE2, see Figure 5 except that the region of stability is above the curve and theregion of instability is below.

However, if we take a pioneer fitness with positive concavity, we can producean example where stocking is required to restabilize E2. When f! (y;) > 0, wesee from (4.12) that we add the following negative term to (6.1) to obtain thenumerator of aGlaa:

-4C12c2J;I (YT)(YTC22-Y;C12). (6.3)Consider exponential fitnesses for both the pioneer and the climax populations

of the forms:

fi64 = exp(r - rvJ and hcv2) = Ysw - 2Y2h (6.4)Here r > 0 is the pioneer decay rate and measures the decrease in pioneer per cap-ita growth with respect to an increase in weighted total density at equilibrium.

For these fitness functions, y; = y; = 1 and we find that f;(y;) = -r,

f;’ (Y;> = r2, and MYI )5= -1. We assume that cl2 = c2r = 1 and c22 = 5 . As r

increases, the term in (6.3) becomes more dominant in the numerator of aG/&z.9For instance, if r = 3 then the period-doubling bifurcation occurs at &r = ?

whereE = z ’2( )

17, 17 . Adding (6.1) and (6.3) and dividing by (6.2) we obtain

dG -47- = -*aa 51

Thus for this pioneer fitness the slope of the bifurcation curve is positive, andharvesting is needed to restabilize E2 as with the linear pioneer above. Hywegver,if r = 6 the period doubling-bifurcation occurs at err = 3 where E2 = ( >ii'E *We compute

aG l4 and aG - 4aa=z %l i i ’

Thus the slope of this bifurcation curve is -7/6. Hence E, loses stability as err

decreases through 211 = 3 and stability is restored by increasing a enough so thatthe system corresponds to a point in parameter space above and to the right of thebifurcation curve (see Figure 6). Numerical experiments show that the destabiliz-


ing effect of decreasing err below 3 by increments of 0.01 may be compensatedfor by increasing a by increments of 0.01, which is consistent with our calcula-

tion of - 716 as the slope of the bifurcation curve.If we study the position of the isoclines near the equilibrium E, (see Figure 7)

we see that increasing err swings the pioneer isocline to the left and moves E2 upthe climax isocline and decreasing cl1 swings the pioneer isocline to the right andmoves E, down the climax isocline. In case (iv) decreasing cl1 moves E2 downand destabilizes it. Our arguments show that stocking or harvesting may berequired to restabilize E2 depending on the concavity of the pioneer fitness. If theconcavity is large then stocking is needed, which causes E, to move farther downthe climax isocline and to attain a higher pioneer density. If the concavity is smallthen harvesting is needed, which causes E, to move back up the climax isoclineand to reduce pioneer density. However, E, may require stocking to restabilizeeven with zero concavity as shown by our next example which satisfies (iii).For our last example, we take the interaction between a pioneer and a climaxpopulation which occurs when the pioneer fitness is linear and the climax fitnessis quadratic:

.UY~=~-Y~ and .MYJ=~Yz-Y~ -2. (6.5)With these fitnesses y; = 1, yi = 3, f;(y;) = -1, and f;(yl) = -2. We assumethat cl2 = c2r = 1 and cz2 = 3.6. The inequality conditions in (iii) hold3rrd5theperiod-doubling bifurcation occurs at &t = g = 0.34848 where E, =

( 1-,- .

Selgrade and Roberds [19] prove that a period-doubling bifurcation occ:; i’c,,increases through &r giving rise to a stable 2-cycle. In fact, as err continues toincrease, a cascade of period-doubling bifurcation occurs and culminates in astrange attractor with two connected components when cl1 = 0.391. Here we con-sider system (SH) as a two parameter bifurcation problem in a and err. Using(4.9) and (4.12), we find that the slope of the bifurcation curve is approximately0.41736 at (a, &r) = (0, $) . Hence stocking is needed to reverse the firstperiod-doubling and to restabilize the equilibrium. In fact, numerical studiesindicate that each bifurcation in the cascade may be reversed by an appropriatelevel of stocking. If cl1 is fixed at 0.391 and a increases from 0 to 0.11 then thestrange attractor is transformed back to a stable equilibrium. Figure 8 depictsperiod-halving which occurs as a increases from a = 0.0026 to a = 0.006 - anattracting 16-cycle bifurcates to an g-cycle and then to a 4-cycle.


UNSTABLE

STABLE

a

FIGURE 6 Bifurcation curve near (0, ztt ) for exponential pioneer (6.4) with r = 6, which showsthat stocking restabilizes E,

\P

\P

-2

X

\

\

. .

2

FIGURE 7 Pioneer (p) and climax (c) isoclines near E2 for a a 0


FIGURE 8 Period-halving for cI1 = 0.391 and 0.0026 s a L 0.006 and fitnesses (6.5)

7. SUMMARY

Here we derive four sets of inequality conditions (see (3.10), (3.12), (3.14), and(3.15)) which guarantee that an equilibrium loses stability through a period-dou-bling bifurcation with respect to the intrinsic pioneer crowding parameter cu. Weintroduce into the system a stocking or harvesting term which is proportional tothe pioneer density. The constant of proportionality is the parameter a. The char-acter of the bifurcation curve in the (a&-parameter space near the point(0, &I), where the equilibrium loses stability, determines whether stocking orharvesting restabilizes the equilibrium. In the case of the equilibrium E, ofprey-predator type, if the pioneer fitness has nonnegative concavity then E, isrestabilized if and only if stocking is done. For the equilibrium E, of competitivetype, we exhibit examples which show that the size of the concavity of the pio-neer fitness function determines whether stocking or harvesting is needed torestabilize E,. Finally we discuss an example where a period-doubling cascademay be reversed and maintained at any attracting cycle along the cascade byappropriate levels of pioneer stocking.

1

.

AcknowledgementThe author thanks James Roberds for helpful conversations.


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Documents

Using Stocking or Harvesting to Reverse Period-Doubling ... · competition and predation in different regions of the same phase space. Yakubu [16] shows that stocking or harvesting